Use zero through fourthorder Taylor series expansions to pre

Use zero- through fourth-order Taylor series expansions to predict f(2) for f(x) = In x using a base point at x = 1. Compute the true percent relative error e, for each approximation. Discuss the meaning of the results.

Solution

clc
clear all

syms a
f(a) = log(a);
df1(a) = diff(f,a);
df2(a) = diff(df1,a);
df3(a) = diff(df2,a);
df4(a) = diff(df3,a);

a = 1;
y1=feval(f,a);
y2=feval(df1,a);
y3=feval(df2,a);
y4=feval(df3,a);
y5=feval(df4,a);

x = 2;
y_approx0 = y1
y_approx1 = y1+(x-a)*y2
y_approx2 = y1+(x-a)*y2+(((x-a)^2)/2)*y3
y_approx3 = y1+(x-a)*y2+(((x-a)^2)/2)*y3+(((x-a)^3)/6)*y4
y_approx4 = y1+(x-a)*y2+(((x-a)^2)/2)*y3+(((x-a)^3)/6)*y4+(((x-a)^4)/24)*y5

y_exact = log(2);

error0 = (y_exact - y_approx0)*100/y_exact
error1 = (y_exact - y_approx1)*100/y_exact
error2 = (y_exact - y_approx2)*100/y_exact
error3 = (y_exact - y_approx3)*100/y_exact
error4 = (y_exact - y_approx4)*100/y_exact